The third and fourth pluricanonical maps of threefolds of general type

For a nonsingular projective threefold of general typeXover the field of complex numbers, we show that the fourth pluricanonical map ϕ4is not birational onto its image if and only ifXis birationally fibred by (1,2)-surfaces, provided that vol(X) ≥ 303. We also have similar characterization of birationality of ϕ3.

Pluricanonical systems for 3-folds and 4-folds of general type

We explicitly find lower bounds on the volume of threefolds and fourfolds of general type in order to have non-vanishing of pluricanonical systems and birationality of pluricanonical maps. In the case of threefolds of large volume, we also give necessary and sufficient conditions for the fourth canonical map to be birational.